Keywords: cable networks, form finding, force density, Pareto, NSGA II.

In this paper an application of a multiobjective optimization scheme using genetic algorithms into the form finding and sizing optimization procedure of tension structures is proposed. Form finding is achieved through the implementation of the force density method introduced by Schek [1]. The NSGA II [2] algorithm is introduced for the optimization process.

The term "form-finding" signifies a procedure whose goal is to predefine geometric configurations of cable nets under a specified loading in such a way that equilibrium is satisfied throughout the structure. Usually the initial shape of the structure that corresponds to service loads is generated. Various methods of form finding have been suggested such as the force density method [1], the grid method [3] and the method of dynamic relaxation [4]. The force density method is a versatile procedure for the derivation of possible equilibrium shapes. Its main advantage over the others is that it transforms the nonlinear problem of form-finding into a linear one. Even so, form-finding can become a tedious task, as the engineer has to take into account various conflicting objectives. These include variations of pretension and different loading scenarios, which alter the final shape of the cable structure.

To overcome such difficulties, variations of the force-density method have been proposed. First Schek [1] introduced the non-linear force density method by imposing geometrical and stress constraints into the form-finding process. He also proposed a damped least squares algorithm for the solution of the optimization problem which nevertheless needs an initial equilibrium shape close to the desired one. Tibert [5] argues that the nonlinear force density method has a slow convergence ratio especially in structures with highly distorted meshes. Kanno and Ohsaki [6], proposed a procedure based on the complementarity of virtual work. Their method of form-finding introduces a second order optimization algorithm in an attempt to circumvent the problems of the nonlinear force density method. Genetic Algorithms have been implemented by the authors [7] into the form finding procedure of cable networks as a single objective optimization procedure.

Since the post tensioning applied to the members, and their residual strength are competitive in nature, no "optimal" solution can exist. One can only search for combinations of post tensioning and weight that are all neutral to each other, in the sense that one cannot minimize the weight without further deviating from the target configuration and vice versa.

1

H.J. Scheck, "The force density Method for form finding and computation of general networks", Computer Methods in Applied Mechanics and Engineering, 3, 115-134, 1974 doi:10.1016/0045-7825(74)90045-0

2

K. Deb, A. Pratap, S. Aqarwal, T. Meyarivan, "A Fasr and Elitist Multiobjective Genetic Algorithm NSGA - II", IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 182 - 197, 2002. doi:10.1109/4235.996017

3

A. Siev, J. Eidelman, "Stress analysis of prestressed suspended roofs", J. Struct. Eng. Division, 47(4/5), 673-682, 1964.

4

M.R. Barnes, "Form finding and analysis of prestressed nets and membranes", Computers & Structures, 30, 685-695, 1988. doi:10.1016/0045-7949(88)90304-5

5

G. Tibert, "Numerical Analysis of Cable Roof Structures", Msc. Thesis, Royal Institute of Technology, Dep. Of Structural Engineering, Stockholm, 1999.

6

Y. Kanno, M. Ohsaki, "Minimum principle of complementary energy of cable networks by using second-order cone programming", J. Solids and Structures, 40, 4437-4460, 2003. doi:10.1016/S0020-7683(03)00215-4

7

S. Triantafyllou, "Nonlinear Analysis and Structural Optimization of Cable Strucutres", Msc Thesis, National Technical University of Athens, Dep. Of Civ. Eng., Athens, 2007.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £145 +P&P)